Simply put, the evolution of the foundational definitions in Algebraic Geometry was driven not by the search of new objects, but of a better understanding of classical objects. In fact, the crowning achievement of Grothendieck's school was precisely to rigorously state and prove a myriad of classical problems. The exemplary case are the Weil conjectures. Weil himself had a dab at the foundations of Algebraic Geometry so as to justify his work on the conjectures. Eventually the problem was closed first by Grothendieck's reinterpretation in terms of étale cohomology, and Deligne's proof of that version.
The starting point is, of course, the study of the zero set of some polynomials. This goes back to Descartes, and his discovery (invention?) of analytic geometry. Now, as time goes on, it is clear that this works well not for real polynomials, but for complex ones. More generally, given an algebraically closed field $k=\overline{k}$, you have a tight relationship between the algebra of the polynomial ring $R:=k[x_1,...,x_n]$ and the geometry of the zero sets in $\mathbb{A}^n_k$ (Hilbert's Nullstellensatz). In particular, these zero sets are found to be in correspondence to radical ideals. In a first iteration, then, an (affine) variety is the zero locus of a radical ideal $I=\sqrt{I}$. A few addendums are in order: (a) one quickly notes that the algebra of the coordinate ring $R/I$ distinguishes between irreducible components determined by minimal primes (Scheinnullstellensatz). Just as topology naturally focuses on connected spaces, one can focus on irreducible varieties; one will then speak of 'algebraic set' for radical ideals, and 'varieties' for prime ones. (b) a natural variation is to consider also open subsets of varieties, the quasi-affine varieties (c) with the discovery projective space, all this discussion can be reproduced for homogeneous ideals and (quasi)projective verieties. A very good reference for this very classical point of view is the first chapter of Hartshorne's Algebraic Geometry.
The original fauna is, then, one unified more by a heuristic and formal similarity rather than a unified conceptual framework. Quasi-affine varities are not affine, and quasi-projective is not projective. (The counterexample for both negative statements is just the affine plane without the origin.) It is true that (quasi)affine is quasi-projective, but there are great advantages is considering the affine structure of a variety. One way to unify these concepts is by defining a variety as the glueing of (a finite number of) affine varieties. It is, of course, a strictly more general definition, but this was all for the better: the Jacobian variety wasn't originally known to be projective, but it did have affine charts. (In fact, I have the vague notion that Weil introduced this definition precisely because of the Jacobian variety.) A complication deriving from this general methodology is the possibility of glueing things 'wrong', the affine line with double origin being the standard example. To prevent this, we add a separation axiom that is formally the same as the Hausdorff axiom: the diagonal $X\hookrightarrow X\times X$ is closed. (Take a pause to ponder why this does not imply that a variety is Hausdorff.)
As things progress, it becomes clear that the local rings should be part of the data of a variety. Varieties are defined as a topological set of zeros, but there is a gulf between what one thinks of a morphism of varieties, and a general continuous morphism; local rings allow us to make precise the distinction is a way that is less awkward than the more classical definition (see Hartshorne Chapter I Exs.3.2 and 3.3, for example). The difficulty here is how to organize to organize all the data. Thankfully, sheaf theory was there to help and lead Serre to his definition: a variety is a locally ringed space that is locally isomorphic to an affine variety (as a locally ringed space). The separation axiom is of course still imposed.
One could draw a direct line from here to schemes, and in fact most people do that. Whereas the connection is indisputable, there really is a piece of the puzzle missing. All that we said above applies to algebraically closed fields. But if we try to extend any of it to more general fields, thing go awry pretty fast. The standard example is the ideal generated by $x^2+1$ in $\mathbb{R}[x]$, which doesn't correspond to any point in the real line, despite the fact that it generates a prime (indeed maximal) ideal. More generally, any of the polynomials $x^n+x^{n-1}+...+1$ for $n$ even are irreducible, but still don't correspond to any real points. They do correspond, of course, to the complex $n$th-roots of unity. Essentially, for non-algebraically closed fields, 'varieties' are somehow determined by points in larger fields. But mind you that in the example of $n$th roots we deal only with algebraic extensions. In general, if we want to persist in the idea of fixing a large field $K$ over which to o geometry over $k$, we need to go beyond the algebraic closure into some extension of infinite trascendence degree. Weil worked precisely along these lines in his Foundations of Algebraic Geometry. An alternative is not to fix any such (incredibly) large field, but rather to try to keep track of the points over an arbitrary field. It may sound dizzying to do that, but there was at hand a means to do that just at the right moment: we can see the variety as determining a functor from the category of fields to the category of sets, assigning to each fields the points of the variety defined over it.
We finally get to Grothendieck and his school. It turns out that the previous two points are in close connection with each other, especially when we realize that we can keep track of points of a variety over an arbitrary $k$-algebra. The connection here are schemes, which on one hand are locally ringed spaces locally isomorphic to a spectrum of a ring; but on the other, are representatives of functors which obey a descent condition, and admit an atlas. Under this definition, we not only unify the whole set of disparate examples (points 1 and 2), making precise the difference with topology (point 3), but include even the case of non-algebraically closed fields (in fact, any base ring!). The last point is an important one: the scheme $X_\mathbb{R}$ corresponding to $x^2+1$ over $\mathbb{R}$ has one (closed) point, and the local ring of that point has residue field $\mathbb{C}$, precisely the field over which the polynomial splits. Over $\mathbb{C}$, of course, $X_\mathbb{C}$ has two closed points, with residue fields $\mathbb{C}$. There is a map $X_\mathbb{C}\to X_\mathbb{R}$ that is a Galois cover. On the other hand, if $X_\mathbb{R}$ had any points defined over $\mathbb{R}$, they'd show up as fixed points of the Galois action in $X_\mathbb{C}$ (if we'd taken $x^3-1$, for example, there would be in both $X_\mathbb{R}$ and $X_\mathbb{C}$ an extra point corresponding to $1$). In this very concrete way, the scheme keeps track of all points over all fields, while still distinguishing which come from where. I am belaboring the point because it is perhaps the most elementary, but still powerful illustration of how the power of schemes is not so much in enlarging our set of objects, but in clarifying the framework for classical ones. The elegance of the framework only grows as we dig deeper into the subject, with its concise definitions and simpler proofs (compare completeness to projectivity, for example).
That said, speaking of schemes does enlarge our fauna, and by a lot. One obvious examples are non-reduced schemes. But there's much to say on the use of even those non-classical schemes as tools in the study of classical ones. The canonical example: $k[\epsilon]/(\epsilon^2)$ allows us to define (Zarisky) tangent spaces; more generally, Artinian rings are the foundation of deformation theory. On the other hand, these new animals do show up naturally (as some moduli spaces, for example). One important side note here is the existence of primary decomposition, which expresses a scheme as a union of integral schemes and some embedded components with only nilpotents.
After all this, it is fair to ask whither are varieties gone. Elegant and useful as the framework may be, we seem to have lost track of them. This gives us our final definition (so-far?): a variety if a finite-type integral scheme over an (arbitrary) field. Indeed, there is a functor embedding the category of varieties in the category of schemes which makes this precise. In the algebraically closed field, this is in Hartshorne's book; for more general fields, it is in Demazure-Gabriel's.
I seem to have only said standard and well-known facts, so I am not sure that I addressed your question very well. And of course, this should not be taken as a historically accurate description. But just as writing it down helped me put some order in my head, I hope it helps you in some measure.
Best Answer
Yes, your question is closely related to rational varieties.
As the article you link to and Cass' comment explain, a rational variety $X$ of dimension $n$ is one that has a dense open subset that is isomorphic to an open subset of affine space $\mathbf A^n$. This isomorphism allows us to identify the coordinate functions on $X$ with rational functions on $\mathbf A^n$, just as in your example. So this gives a rational parametrisation of the points on $X$.
Conversely, if you have a rational parametrisation of the points of $X$, plugging in values of the parameters gives a dominant rational map $f:\mathbf A^m \dashrightarrow X$. (We don't demand that parametrisations be defined for all possible values of the parameters, just a dense open set of the possible values. Note also that our parametrising affine space is allowed to have bigger dimension than $X$.)
A variety $X$ for which one can find a map $f$ as described above is called unirational. It's an easy exercise to see that if there is a dominant rational map as above, then one can find a dominant map $\varphi :\mathbf A^n \dashrightarrow X$ from an affine space of the same dimension as $X$. Such a map must be generically finite, so it gives a finite-to-one parametrisation of the points of $X$.
However: unirational does not imply rational! It was known by around 1900 that a curve or surface that has a finite-to-one parametrisation must actually have a generically one-to-one (in other words birational) parametrisation. The question of whether this remains true in higher dimensions remained open until the early 1970s, when, remarkably, three completely independent proofs (by Artin–Mumford, Clemens–Griffiths, and Iskovskikh–Manin) showed that a smooth cubic hypersurface in $\mathbf P^4$ is unirational but not rational.
In general, deciding whether varieties are rational or not remains one of the most fundamental and difficult issues in algebraic geometry.
On the other hand: although there are some cases in which deciding rationality is extremely difficult, the comment of Cass that varieties are "rarely" rational is quite accurate, because there are easliy-verified necessary conditions for rationality that rule out "most" candidates.
For example, let $X$ be a smooth hypersurface in $\mathbf P^n$. Then the adjunction formula shows that if $X$ has degree at least $n+1$, then the canonical bundle $\omega_X = \wedge^{n-1} \Omega_X$ has nonzero global sections. This is impossible for a rational variety, and so we conclude that no smooth hypersurface of degree at least $n+1$ is rational.